Respuesta :
- The next four terms in the sequence = [tex]\frac{-5}{8} , \frac{6}{9} ,\frac{-7}{10}[/tex][tex], \frac{8}{11}[/tex]
- The explicit equation for f (n) to represent the sequence, [tex]f(n)=(-1)^{n} \frac{n}{n+3}[/tex]
- The sign of f(59) is negative
Given,
The sequence: [tex]\frac{-1}{4} ,\frac{2}{5} ,\frac{-3}{6} ,\frac{4}{7} ...[/tex]
Part A: We have to find the next four terms in the sequence.
We can use the equation [tex]f(n)=(-1)^{n} \frac{n}{n+3}[/tex]
Where, n is the number of term.
For first term [tex]f(1)=(-1)^{1} \frac{1}{1+3} =(-1)\frac{1}{4} =\frac{-1}{4}[/tex]
Now, we can determine the next four terms in the sequence.
[tex]f(5)=(-1)^{5} \frac{5}{5+3} =(-1)\frac{5}{8} =\frac{-5}{8}[/tex]
[tex]f(6)=(-1)^{6} \frac{6}{6+3} =(1)\frac{6}{9} =\frac{6}{9}[/tex]
[tex]f(7)=(-1)^{7} \frac{7}{7+3} =(-1)\frac{7}{10} =\frac{-7}{10}[/tex]
[tex]f(8)=(-1)^{8} \frac{8}{8+3} =(1)\frac{8}{11} =\frac{8}{11}[/tex]
Part B: The explicit equation for f (n) to represent the sequence.
[tex]f(n)=(-1)^{n} \frac{n}{n+3}[/tex]
Where n is the number of term
Part C: The sign of f(59)
[tex]f(59)=(-1)^{59} \frac{59}{59+3} =(-1)\frac{59}{62} =\frac{-59}{62}[/tex]
From the above calculation it is clear that the sign of f(59) is negative.
Learn more about explicit equation here: https://brainly.com/question/27880277
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